The ideal relativistic rotating gas as a perfect fluid with spin

Understand

We show that the ideal relativistic spinning gas at complete thermodynamical equilibrium is a fluid with a non-vanishing spin density tensor \sigma_\mu \nu.

• After having obtained the expression of the local spin-dependent phase space density f(x,p)_(\sigma \tau) in the Boltzmann approximation, we derive the spin density tensor and show that it is proportional to the acceleration tensor Omega_\mu \nu constructed with the Frenet-Serret tetrad.
• We recover the proper generalization of the fundamental thermodynamical relation, involving an additional term -(1/2) \Omega_\mu \nu \sigma^\mu \nu.
• We also show that the spin density tensor has a non-vanishing projection onto the four-velocity field, i.e.